Assertion (A) :f(x) =[x] is not differentiable at x=2.
Reason (R ) f(x)=[x] is not continuous at x=2.

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is true

Answer

To solve the problem, we need to analyze the assertion and the reason provided regarding the function \( f(x) = [x] \), where \( [x] \) denotes the greatest integer function (also known as the floor function).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = [x] \) gives the greatest integer less than or equal to \( x \). For example:
- \( [1.5] = 1 \)
- \( [2] = 2 \)
- \( [2.5] = 2 \)
- \( [3] = 3 \)

2. **Identifying Points of Interest**:
We are particularly interested in the point \( x = 2 \). We need to check the behavior of the function around this point.

3. **Evaluating Continuity at \( x = 2 \)**:
To check continuity at \( x = 2 \), we need to find the left-hand limit and the right-hand limit as \( x \) approaches 2.

- **Left-hand limit**:
\[ \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} [x] = [2] = 2 \]
- **Right-hand limit**:
\[ \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} [x] = [2] = 2 \]

- **Function value at \( x = 2 \)**:
\[ f(2) = [2] = 2 \]

Since both limits and the function value at \( x = 2 \) are equal, we conclude that \( f(x) \) is continuous at \( x = 2 \).

4. **Evaluating Differentiability at \( x = 2 \)**:
Next, we check if \( f(x) \) is differentiable at \( x = 2 \). For differentiability, we need to check the derivative from both sides.

- **Derivative from the left**:
\[ f'(x) = 0 \quad \text{for } x < 2 \]
- **Derivative from the right**:
\[ f'(x) = 0 \quad \text{for } x > 2 \]

However, at \( x = 2 \), the function \( f(x) \) has a jump discontinuity. The left-hand derivative and right-hand derivative do not exist at this point because the function has a sudden change in value.

5. **Conclusion**:
Since \( f(x) = [x] \) is not differentiable at \( x = 2 \) due to the jump discontinuity, we can conclude that:
- Assertion (A) is true: \( f(x) = [x] \) is not differentiable at \( x = 2 \).
- Reason (R) is false: \( f(x) = [x] \) is continuous at \( x = 2 \).

### Final Answer:
- Assertion (A) is true.
- Reason (R) is false.